“So, any “coin” is always produced by a government?”
I believe so; or else, they are commissioned by the government and minted with government approval. From looking at the site where my wife and I have been buying gold and silver, I noted that “coins” seem to be minted with government approval regardless of the country of origin while anything not approved is called a “round”–as ViennaSausage mentioned above.
PS: ViennaSausage, I replied to your question, albeit belatedly.